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  • Letter
  • Open Access

Winding topology of multifold exceptional points

Tsuneya Yoshida1,2,*, J. Lukas K. König3,†, Lukas Rødland3,‡, Emil J. Bergholtz3,§, and Marcus Stålhammar4,5,6,∥

  • *Contact author: yoshida.tsuneya.2z@kyoto-u.ac.jp
  • †Contact author: lukas.konig@fysik.su.se
  • ‡Contact author: lukas.rodland@fysik.su.se
  • §Contact author: emil.bergholtz@fysik.su.se
  • ∥Contact author: marcus.backlund@physics.uu.se
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Phys. Rev. Research 7, L012021 – Published 24 January, 2025

DOI: https://doi.org/10.1103/PhysRevResearch.7.L012021

Abstract

Despite their ubiquity, a systematic classification of multifold exceptional points, n-fold spectral degeneracies (EPns), remains a significant unsolved problem. In this article, we characterize the Abelian eigenvalue topology of generic EPns and symmetry-protected EPns for arbitrary n. The former and the latter emerge in (2n−2)- and (n−1)-dimensional parameter spaces, respectively. By introducing topological invariants called resultant winding numbers, we elucidate that these EPns are stable due to topology of a map from a base space (momentum or parameter space) to a sphere defined by resultants. In a D-dimensional parameter space (D≥c), the resultant winding numbers topologically characterize (D−c)-dimensional manifolds of generic (symmetry-protected) EPns, whose codimension is c=2n−2 (c=n−1). Our framework implies fundamental doubling theorems for both generic EPns and symmetry-protected EPns in n-band models.

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Phys. Rev. Research 7, L012021– Published 24 January, 2025
  • Received 21 September 2024
  • Accepted 23 December 2024
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